Problem 93 - College Algebra

Question

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{1-\frac{3}{x+2}}{1+\frac{1}{x-2}}$$

Step-by-Step Solution

Answer

The simplified equation for function \(f\) is \(f(x) = \frac{x^2-3x+2}{x^2+4x-4}\). A graph of the function would display its behavior based on this equation

Step 1: Simplify the Complex Fraction

Begin by simplifying the complex fraction. To do this, multiply every term by the common denominator, which is \((x+2)(x-2)\). This yields: \[f(x) = \frac{(x+2)(x-2)-3(x-2)}{(x+2)(x-2)+(x+2)} \]

Step 2: Simplify the Numerator and Denominator

Next, distribute the terms in the numerator and denominator, simplify and collect like terms. This results in:\[f(x) = \frac{x^2-4-3x+6}{x^2+4x-4}\]and further simplifies to\[f(x) = \frac{x^2-3x+2}{x^2+4x-4}\]

Step 3: Graphing the Function

Now that the function has been obtained, plot the function by setting up a graph and determine its behavior based on the function's equation. Note: the graphing step requires numerical ability and understanding of how the function's equation affects its graphical representation. Therefore, this step is better executed using graphing tool or technology.